Gaussian-type density bounds for solutions to multidimensional backward SDEs and application to gene expression

We obtain upper and lower Gaussian-type bounds on the density of each component $Y^i_t$ of the solution $Y_t$ to a multidimensional non-Markovian backward SDE. Our approach is based on the Nourdin-Viens formula and a stochastic version of Wazewski's theorem on the positivity of the components of a solution to an ODE. Furthermore, we apply our results to stochastic gene expression; namely, we estimate the density of the law of the amount of protein generated by a gene in a gene regulatory network.